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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 43.
THE LOGIC OF CLASSES.
We have to distinguish between the subsumptive
use of a word, and its (say) 'nominal'
use. In the former case, the word has an only incidental role, serving to direct
our minds to the objects we label by it; in the latter case, the word itself is
the object of our attention, while the things it refers to are incidental. Thus,
for example, when we speak of dogs, we think of our tail-wagging and barking
friends; but when we speak of "dogs", we mean the word enclosed by
inverted commas itself.
This should not be confused with the distinction between denotative and
connotative terms, which we made in discussing permutation. We can take
denotative terms subsumptively or nominally (as we did with above, comparing
dogs and "dogs"), and we can take connotative terms likewise
subsumptively or nominally (for example, caninity and "caninity").
That is not what is at issue here. What is at issue is, whether our focus is
purely objective (as in, dogs and caninity), or we are focusing on the
instrument (as in, "dogs" and "caninity").
Thus, in subsumptive intent, we mean what the word refers to, and the
instrument is transparent; whereas in nominal intent, the instrument itself is
what we mean, and what the word refers to specifically is of lesser moment.
Normally, our intent is subsumptive (let us symbolize this as an X); but
sometimes, especially in epistemological discussions, our intent is nominal (let
us symbolize this as "X").
Nomenclatural propositions have the primary forms: "X" is the
name of all X; or: all X are the referents of "X".
We may extend the distinction between subsumptive and nominal intent to
other aspects of our instruments of thought, not only to the verbal. An 'idea'
or a 'class', or any similar construct, may like a word be considered
'nominally', in contrast to subsumptively. This does not mean to imply that
words, ideas, classes, and such, are all one and the same thing, but only that
they have in common the property we mentioned. There is no doubt that they are
significantly different concepts, yet also somehow related.
The precise relation between these various concepts is not the topic of
this chapter. Rather, we shall specifically explore some of the mechanics of
classification, in an effort to better understand the logical relations between
things and our concepts of them. This research into the
way knowledge is organized, has been of great interest to logicians of this
century, under the heading of 'the logic of classes'.
We think of a class and its members, as having a similar relation to that
of a receptacle and the things it contains. The container, an elastic and
permeable wrapping, is a figment of our imagination; yet, its shape and size are
determined by the contents. This visual analogy is not perfect, but is a
starting point.
The subsumptive outlook is directed at the contents, labeling each member
as X; this is the only kind of classificatory relationship we have dealt with so
far: it is the concern of Aristotelean logic. The nominal outlook is directed at
the container, labeling the class as "X"; this gives rise to a new
field of logic.
a.
Definitions.
For any thing X, we can invent a corresponding thing "X", such
that: ·
whatever is X, is a member of
"X"; and whatever is not X, is not a member of "X".
Conversely, we say: ·
"X" is the class of (all)
X; and "X" is a class of anything which is X, and not a class of
anything which is not X.
These two sets of statements mean the same thing, they are just two sides
of the same coin, they commute each to the other; we call the whole relationship
'class-membership'.
The above begin to formally define the difference between what we mean by
X and "X", relating them through a new pair of copulas, which are
different from the copula 'is'. In one direction, the copula is labeled 'is a
member of', and has a subsumptive as subject and a nominal as predicate; in the
other direction, the copula is labeled 'is a class of', and has a nominal as
subject and a subsumptive as predicate.
Since in speech, unlike writing, we have no way to display inverted
commas, we merge the two and say: this thing is a member of the class of X; the
latter expression, class of X, is equivalent to "X", in relation to X.
We understand "X", or the class of X, as a mental construct of some
sort, which we intend to bear a certain relation to the things we have labeled
as X. We assign the virtually same label to the construct as we did to the
original things, except for a small distinguishing mark ("" or the
class of) to keep their distinction in mind.
The plain name is subsumptive, referring directly to the things
concerned, the marked name is nominal, referring rather to the invented
correspondent of the things. Note that, although the member to class relation
has some similarity to the relation of an individual to a group, they are not
identical. The subsumptive versus nominal distinction, should not be confused
with the dispensive versus collective (or even collectional) distinction, which
we made earlier (in the discussion of quantity).
Thus far, what we have done is to point to a set of phenomena, which we
commonly encounter in our current ways of thinking, and sorted them out
somewhat, and named the various factors. But all we have achieved is at best a
technical definition; a fuller definition requires some further understanding of
the distinctive properties of these factors. That is what we will look into now.
Consider the following example, which accords with our normal manner of
speaking. Dogs are dogs, and are members of "dogs" (or the class of
dogs). But, dogs are not "dogs", only members of "dogs"; and
"dogs" is not a dog, and not a member of "dogs". Note that
there is no self-contradiction in saying that dogs are not "dogs", or
that "dogs" is not a dog, even though the statement that dogs are not
dogs is of course absurd.
Such examples suggest the following features and processes. (Note that I
concentrate mainly on the properties of 'is a member of'; those of 'is a class
of' follow obviously, I do not highlight them, to avoid repetitions.)
b.
Features.
Whereas in a proposition of the form 'this thing is X', the subject and
predicate are both subsumptive — in a proposition of the form 'this thing is a
member of "X" (or the class of X)', the predicate is nominal. This principle is necessary, because
the whole concept of membership was built with the intent to study that special
kind of term we call nominal. Membership by definition relates any kind of thing
to one kind of thing specifically: mental constructs.
With regard to the subject of membership, the above definition concerns
only subsumptive subjects, but we shall presently consider nominal ones.
What is X, is not "X",
but only a member of "X". The copula 'is a member of' positively
relates two things, X and "X", which the copula 'is' negatively
relates, at least in examples like ours (dogs are not "dogs", but only
members of "dogs").
"X" is not an X, nor a member of "X". A class is not a member of itself:
the relation of membership is not reversible, at least not in examples like ours
("dogs" is not a member of "dogs", since it is not a dog).
With regard to the latter two principles, the examples only prove that
they hold in some instances; however, we may generalize from such cases, if we
find no examples to the contrary.
c.
Immediate inferences.
Obviously, by definition, since all X are X, all X are members of
"X"; and since no nonX are X, no nonX are members of "X".
The class of X includes all things which are X, and excludes all things which
are not X. Similar eductions apply for the class of nonX, or "nonX".
It follows that membership in "X" and membership in "nonX"
are exact contradictories.
More broadly, we can infer from the above definition of membership that: if
any X is Y, that X is a member of "Y"; and if any X is not Y, that X
is not a member of "Y". Any thing which is X and also Y, is an
X which is a member of "Y"; any thing which is X but not Y, is an X
which is not a member of "Y". That is, "X" is the
class of all X, but not the only class
for any X; there are normally other classes like "Y", of which we can
say that it is a class of some or all
X. For examples, retrievers are members of the class of dogs, but not members of
the class of cats.
It follows that: if all X are Y, then all X are members of "Y";
if only some X are Y, then only some X are members of "Y"; if no X is
Y, no X is a member of "Y". (Note in passing, in the middle case, we
regard the membership of some X in "Y", as 'accidental', or
'incidental' to their being X, since not all X fall in this category; or we say
that these X are Y, but not 'qua' X or
not as X 'per se', not by virtue of
being X.)
These statements are reversible: if all X are members of "Y",
then all X are Y; if some X are members of "Y", then some X are Y; if
some X are not members of "Y", then some X are not Y; if no X are
members of "Y", then no X are Y.
d.
Deductive arguments.
We thus can construct the following syllogisms for the copulas 'is (or is
not) a member of', on the basis of Aristotelean syllogisms for the copulas 'is
(or is not)'.
Figure 1.
All Y are members of "Z",
all/this/some X is/are member(s) of "Y",
so, all/this/some X is/are member(s) of "Z".
Likewise, with negative major and conclusion.
Figure 2.
No Z are members of "Y",
all/this/some X is/are member(s) of "Y",
so, all/this/some X is/are not member(s) of "Z".
Likewise, with positive major and negative minor.
Figure 3.
All/this/some Y are members of "Z",
some/this/all Y is/are member(s) of "X",
so, some X are members of "Z".
Likewise, with negative major and conclusion.
Figure 4.
No Z are members of "Y",
some Y are member(s) of "X",
so, some X is/are not members of "Z".
Such deductions are easily validated, by translating them into their
customary forms. Note that a term may be subsumptive in one proposition and
nominal in another, according to its position by virtue of the figure.
e.
Modal class-logic.
The definition of class-membership is easily modalized, if we wish to
work out a more modal class logic.
Thus, for natural modalities: if something can be X, then it can be a
member of "X"; and if something cannot be X, it cannot be a member of
"X"; if something must be X, then it must be a member of
"X"; and if something can not-be X, it can not-be a member of
"X". Similarly for temporal modalities. The quantification of these
singular forms represents extensional modality.
Note that these definitions are in the form of extensional conditionals.
The logical properties of their consequent forms are easily derived from the
modal logic of their antecedent forms, which are ordinary categoricals. That
includes: oppositions, eductions, and deductions.
In the previous section, we defined and analyzed the membership of a
non-class (subsumptive) in a class; now, we need to look into what we mean when
we say of a class (nominal) that is a member of another class.
a.
Definitions.
We propose that, for any X and Y: ·
if all X are Y, then "X"
(or the class of X) is a class of Y, and therefore is a member of "classes
of Y", (or the class of classes of Y).
Conversely: ·
if less than all or no X are Y, then
"X" (or the class of X) is not a class of Y, and therefore not a
member of "classes of Y" (or the class of classes of Y).
Note the variety in wording; we also often abbreviate 'class(es) of Y' to
'Y-class(es)'.
This definition of so-called classes of classes reflects our common
practise. For examples, since all dogs are animals, "dogs" is an
animal-class, or a member of "animal-classes"; but since some dogs are
not black animals, "dogs" is not a class of black animals, or a member
of "classes of black animals".
Now, this is an artifice. The reason why we construct this new concept is
that we want to be able to talk about classes in the same way as we talk about
things. We build up a parallel domain, in which classes bear relations to each
other, somewhat similar to the relations between their ultimate referents. Thus
far, the stratification of things had no equivalent in the realm of classes,
since nominal terms were defined as predicates of the 'is a member of' copula.
In order to place classes as subjects of similar propositions, we introduce
appropriate special predicates: classes of classes. A class of classes is a
subsumptive whose referents are specifically nominal.
Note that an ordinary class (that is, one which is not a class of
classes) stands as subject of membership when the predicate is a class of
classes; there are no grounds for assuming that an ordinary class can ever be a
member of another ordinary class. We cannot, for instance, say "dogs"
is a member of "animals", but only, dogs are members of
"animals", or "dogs" is a member of
"animal-classes".
This was already suggested in the previous section, in the claim that
"X" is not a member of "X"; now, we can generalize further,
and say that "X" cannot be a member of any "Y", granting
that these terms are not classes of classes of anything. Other than the above
defined case, there
are no known conditions regarding any X and Y, under which we could conclude
that "X" is a member of "Y".
Similarly, there are no known conditions
under which propositions of the form: "X-classes" is a member of
"Y", may arise. However, as we shall presently see,
propositions of the form: all/some X-classes are (are not) members of
"Y-classes", do indeed arise, directly out of the definition of
classes of classes. However, note that the subject is subsumptive here, not
nominal.
Let us now investigate how successful our above definition of classes of
classes is, some of the logical properties it implies.
b.
Features.
"X"
is an X-class, and a member of "X-classes" (or the class of
X-classes), since all X are X, and even though "X" is not an X, nor a
member of "X". This principle proceeds deductively from the
definition, by substituting X where we find Y. It means that every class is a
member of the class of classes bearing its name. It does not mean that it is a
member of itself, however; we should not confuse a class with a class of
classes; thus far, we have no cause to doubt the earlier postulate that classes
cannot be members of themselves. For example, "dogs" is a dog-class,
and a member of "dog-classes".
However, no X is an X-class, nor a member
of "X-classes", even though all X are members of
"X", and "X" is a member of "X-classes". The
definition of a class of classes refers to a nominal "X" as its
subject, not a subsumptive X. The relationship of membership is not passed on
all the way down the chain to the individuals subsumed by X; the only
individuals subsumed by a class like "X-classes" are classes like
"X". For example, dogs are members of the class of dogs, but not of
the class of dog-classes.
Similarly, no X is a Y-class, nor a member
of "Y-classes", even if all X are Y, and therefore members of
"Y". Contrast those statements to saying that "X" is a
Y-class (or a class of Y), and therefore a member of "Y-classes" (or
the class of Y-classes, or the class of classes of Y). Keep the distinctions
clear.
We might strengthen these insights by calling ordinary classes, classes 'of
the first order', and classes of classes, classes 'of the second order'; then we can say: members of a class of the first
order cannot be members of a class of the second order; at best, they
might be said to be members of a member of a class of the second order. This may
be referred to as the principle of
intransmissibility of membership across orders of classification.
c.
Immediate Inferences.
Obviously, by definition, if "X" is a Y-class, then all X are
Y; and if "X" is not a Y-class, at least some X are not Y. Likewise,
with any of the alternative wordings.
The following theorems are important, because they construct propositions
in which a class of classes is the subject, a novelty; thus far, classes of
classes only appeared as predicates.
If
all X are Y, then all X-classes (including "X" itself) are Y-classes,
or members of "Y-classes", the class of Y-classes. Proof is by
exposition: consider any class "W" which fits the definition of an
X-class, so that all W are X, then (since all X are Y) all W are Y, and it will
follow that "W" is a Y-class; this can be repeated for any
"W", and even "X" fits in (since all X are X). For example,
all dog-classes (such as "retrievers") are animal-classes.
A corollary is: if "X" is a Y-class,
then all (other) X-classes are (also) Y-classes; the conclusion follows,
since the premise implies that all X are Y.
If
some X are Y, then some X-classes are Y-classes.
Proof: those things which are both X and Y can be said to be XY, and
self-evidently all XY are X and all XY are Y; thus we have, in the case of
"XY" at least, an X-class which is a Y-class.
If
some X are not Y, then some X-classes are not Y-classes.
Proof: those things which are X but not Y can be said to be XnonY, and
self-evidently all XnonY are X and no XnonY are Y; thus we have, in the case of
"XnonY" at least, an X-class which is a not Y-class.
If
no X is Y, then no X-classes are Y-classes. For if, say, "W"
is an X-class, then all W are X; and since no X is Y, it follows that no W is Y,
which means that "W" is not a Y-class.
Thus, note well, if some X are Y, it follows only that some X-classes are
Y-classes, for we may find a class "W" (other than "X") for
which all W are X and yet no W is Y. Likewise, if some X are not Y, it follows
only that some X-classes are not Y-classes, for we may find a class
"W" (other than "X") for which all W are X and also all W
are Y.
Conversely, if all X-classes are Y-classes, then all X are Y; if some
X-classes are Y-classes, then some X are Y; if some X-classes are not Y-classes,
then some X are not Y; and if no X-classes are Y-classes, no X is Y.
d.
Deductive arguments.
It is important to note that syllogistic
reasoning with the copula 'is a member of' depends for its validity on the
manner of reference of its terms.
We saw that, if any X is a member of "Y", and "Y" is
a member of "Z", it follows that that X is a member of "Z".
The proof being, since that X is Y, and all Y are Z, then that X is Z.
However, if even all X are members of "Y", and "Y" is
a member of "Z-classes", it does not follow that any X is a member of
"Z-classes". For, even though it be implied that all X are Z, this
only signifies, as already pointed out, that "X" is a member of
"Z-classes", not that any X is a Z-class.
Thus, we have the same arrangement of premises, with the copula 'is a
member of' in both cases, yet the conclusions are of fundamentally different
form. In the former case, subsumptives are members of an ordinary class; in the
latter case, a nominal is member of a class of classes. This of course
illustrates the earlier mentioned principle of intransmissibility of membership.
The following arguments may be validated with reference to the indicated
Aristotelean syllogisms.
Figure 1 (from 1/AAA).
"Y" is a member of "Z-classes",
and "X" is a member of "Y-classes",
therefore, "X" is a member of "Z-classes".
Figure 2 (from 2/AOO).
"Z" is a member of "Y-classes",
and "X" is not a member of "Y-classes",
therefore, "X" is not a member of "Z-classes".
Figure 3 (from 3/OAO).
"Y" is not a member of "Z-classes",
and "Y" is a member of "X-classes",
therefore, "X" is not a member of "Z-classes".
However, no other arguments of that sort are possible. In the first
figure, a negative major premise, "Y" is not a member of
"Z-classes", would only imply that some Y are not Z, from which no
conclusion can be drawn; and as for 1/AII, it has no equivalent here, since "X" is a member of
"Y-classes" would require that all X be Y. In the second figure,
likewise with regard to a negative major premise; and as for 2/AEE,
it has no equivalent here, since "X" is not a member of
"Y-classes" only implies that some X are not Y. We can similarly write
off the remaining moods of the third figure. The fourth figure has no equivalent
here, since the minor premise of 4/EIO
is not enough to imply membership of a class in a class of classes.
Thus, we only have three valid moods for propositions of this kind; no
other moods are valid. The first is used for including a class in a class of
classes, the other two for purposes of exclusion. These can be restated as
follows, in accordance with the theorems of immediate inference:
Figure 1 (1/AAA)
"Y" is a Z-class (or, all Y-classes are Z-classes),
"X" is a Y-class (or, all X-classes are Y-classes),
so, "X" is a Z-class (or, all X-classes are Z-classes).
Figure 2 (2/AOO).
"Z" is a Y-class (or, all Z-classes are Y-classes),
"X" is not a Y-class (or, some X-classes are not Y-classes),
so, "X" is not a Z-class (or, some X-classes are not
Z-classes).
Figure 3 (3/OAO)
"Y" is not a Z-class (or, some Y-classes are not Z-classes),
"Y" is an X-class (or, all Y-classes are X-classes),
so, "X" is not a Z-class (or, some X-classes are not
Z-classes).
For examples. (i) The class of retrievers is a class of dogs, and the
class of dogs is a class of animals, therefore "retrievers" is an
animals-class. (ii) "Roses" is a class of plants, but "dogs"
is not a class of plants, therefore "dogs" is not a member of
"classes of roses". (iii) "roses" is not a class of animals,
but "roses" is a class of plants, therefore "plants" is not
a member of the class of classes of animals.
Although the subsumptive relation between classes and classes of classes
allows for only these three valid moods, it is clear that the subsumptive
relation of classes of classes with each other allows for a fuller range of
syllogistic argument. The three arguments indicated in brackets are obviously
not all the valid moods for such terms, but any valid Aristotelean syllogism
might be applied here. For example: some X-classes are Y-classes, no Y-classes
are Z-classes, therefore some X-classes are not Z-classes. The explanation is
simply that first order classes are effectively singular, whereas second order
class subsume many such singulars.
e.
Modal class-of-classes logic.
To modalize the concept of classes of classes, we would have to appeal to
a collectional proposition, of the
form 'all X can be Y. This, you may
recall, signifies, not only that for each X there are some circumstances in
which it is Y, but also that there is at least one set of circumstances in which
all X at the same time are Y.
The modal definitions are: for any X and Y, if all
X simultaneously can be Y, then "X" can be a class of Y; but if some X
cannot be Y, or all X can be Y, but not all
at once, then "X" cannot be a class of Y; and if all X must be Y, then
"X" must be a class of Y; but if some X can not-be Y, then
"X" can not-be a class of Y.
From these definitions, the entire modal logic of classes of classes is
easily derived, with reference to the logic of ordinary modal categoricals and
collectionals. Note that the defining propositions are all intended as
extensional conditionals, but two of them are special in that they contain a
collectional antecedent. |
